#8285564

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,10)(3,8,11,6)(4,5,9,15)(12,13,16,14)(17,18,23,24)(19,26,21,25)(20,29,30,22)(27,32,31,28)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(18,24)(20,30)(25,26)(27,31)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(3,11)(6,8)(12,16)(13,14)(17,19,23,21)(18,25,24,26)(20,27,30,31)(22,32,29,28)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,20)(18,22)(19,27)(21,31)(23,30)(24,29)(25,32)(26,28)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,21,23,19)(18,26,24,25)(20,27,30,31)(22,32,29,28)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,30)(25,31)(26,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8192760);